L10n43

(Knotscape image)

See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L10n43's page at the original Knot Atlas.

Planar diagram presentation	X₈₁₉₂ X_10,4,11,3 X_20,10,7,9 X₂₇₃₈ X_4,15,5,16 X_5,13,6,12 X_16,12,17,11 X_17,6,18,1 X_19,15,20,14 X_13,19,14,18
Gauss code	{1, -4, 2, -5, -6, 8}, {4, -1, 3, -2, 7, 6, -10, 9, 5, -7, -8, 10, -9, -3}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 u 2 + 3 v u 2 - u 2 + 2 v 2 u -5 v u + 2 u - v 2 + 3 v -1$ (db)
Jones polynomial	$-q^{13/2}+3 q^{11/2}-4 q^{9/2}+6 q^{7/2}-7 q^{5/2}+6 q^{3/2}-6 \sqrt{q}+\frac{3}{\sqrt{q}}-\frac{2}{q^{3/2}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$z 5 a -3 -3 z 3 a -1 + 3 z 3 a -3 - z 3 a -5 + 2 a z -6 z a -1 + 4 z a -3 - z a -5 + 2 a z -1 -3 a -1 z -1 + a -3 z -1$ (db)
Kauffman polynomial	$- z 8 a -2 - z 8 a -4 -2 z 7 a -1 -5 z 7 a -3 -3 z 7 a -5 - z 6 a -2 -3 z 6 a -4 -3 z 6 a -6 - z 6 + 5 z 5 a -1 + 13 z 5 a -3 + 7 z 5 a -5 - z 5 a -7 + 4 z 4 a -2 + 12 z 4 a -4 + 8 z 4 a -6 -3 a z 3 -12 z 3 a -1 -15 z 3 a -3 -4 z 3 a -5 + 2 z 3 a -7 -7 z 2 a -2 -9 z 2 a -4 -4 z 2 a -6 -2 z 2 + 5 a z + 10 z a -1 + 7 z a -3 + 2 z a -5 + 3 a -2 + a -4 + 3-2 a z -1 -3 a -1 z -1 - a -3 z -1$ (db)

The coefficients of the monomials

t r q j

are shown, along with their alternating sums

χ

(fixed

j

, alternation over

r

). The squares with yellow highlighting are those on the "critical diagonals", where

j -2 r = s + 1

j -2 r = s -1

, where

s =

1 is the signature of L10n43. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = 0$	$i = 2$
$r = -2$	${\mathbb Z}^{2}$	${\mathbb Z}$
$r = -1$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = 0$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{3}$
$r = 1$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 2$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 3$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4}$	${\mathbb Z}^{4}$
$r = 4$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{2}$
$r = 5$	${\mathbb Z}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{2}$
$r = 6$	${\mathbb Z}_2$	${\mathbb Z}$